Special cases of the questions considered in the previous section are known to hold for surfaces that are invariant under a Fuchsian group action — that is, a group action that leaves invariant a plane. This happens when the two metrics \(h_-\)  and \(h_+\)  considered above are identical. Existence and uniqueness results of this type were obtained for the third fundamental form of smooth surfaces in \({\mathbb {H}}^3\)  in [1]}, and for the induced metrics on polyhedral surfaces in e.g. [2]}, [3]}, [4]}, [5]}.
